Goodness of Fit of Social Network Models∗

David R. Hunter, Penn State UniversitySteven M. Goodreau, University of WashingtonMark S. Handcock, University of Washington

September 19, 2006

Abstract

We present a systematic examination of a real network dataset using maxi-mum likelihood estimation for exponential random graph models as well as newprocedures to evaluate how well the models fit the observed networks. Theseprocedures compare structural statistics of the observed network with the corre-sponding statistics on networks simulated from the fitted model. We apply thisapproach to the study of friendship relations among high school students fromthe National Longitudinal Study of Adolescent Health (AddHealth). We focusprimarily on one particular network of 205 nodes, though we also demonstratethat this method may be applied to the largest network in the AddHealthstudy, with 2209 nodes. We argue that several well-studied models in thenetworks literature do not fit these data well, and we demonstrate that thefit improves dramatically when the models include the recently-developed geo-metrically weighted edgewise shared partner (GWESP), geometrically weighteddyadic shared partner (GWDSP), and geometrically weighted degree (GWD)network statistics. We conclude that these models capture aspects of the socialstructure of adolescent friendship relations not represented by previous models.

Key Words: degeneracy, exponential random graph model, maximum likeli-hood estimation, Markov chain Monte Carlo, p-star model

1 Introduction

Among the many statistical methods developed in recent decades for analyzing de-

pendent data, network models are especially useful for dealing with the kinds of

dependence induced by social relations. Applications of social network models are

∗The authors are grateful to Martina Morris for numerous helpful suggestions. This research issupported by Grant DA012831 from NIDA and Grant HD041877 from NICHD.

1

School 10: 205 Students

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Figure 1: Mutual friendships represented as a network. Shapes of nodes denote sex:circles for female, squares for male, and triangles for unknown. Labels denote theunits digit of grade (7 through 12), or “–” for unknown.

becoming important in a number of fields, such as epidemiology, with the emergence

of infectious diseases like AIDS and SARS; business, with the study of “viral mar-

keting”; and political science, with the study of coalition formation dynamics. Much

recent effort has been focused on inference for social network models (e.g., Holland

and Leinhardt, 1981; Strauss and Ikeda, 1990; Snijders, 2002; Hunter and Handcock,

2006), but comparatively little work tests the goodness of fit of the models.

Data on social relationships can often be represented as a network, or mathemati-

cal graph, consisting of a set of nodes and a set of edges, where an edge is an ordered

or unordered pair of nodes. This article focuses specifically on network data collected

at a nationally representative sample of high schools in the United States. The nodes

represent students and the edges signify friendships between pairs of students. Fig-

ure 1 depicts one such network graphically, where the shapes and labels of the nodes

represent covariates measured on the students.

We consider exponential family models, in the traditional statistical sense, for

network structure. These models have a long history in the networks literature, and

we refer to them here as exponential random graph models (ERGMs). The primary

2

contribution of this article is to propose a systematic approach to the assessment of

network ERGMs. The models we examine here achieve a good fit to key structural

properties of the network with a small number of covariates. The approach and the

findings address a central question in the network literature: Can the global structural

features observed in a network be generated by a modest number of local rules?

Another contribution of this paper is to demonstrate the use of maximum like-

lihood to fit reasonable models to network data with hundreds of nodes and obtain

results that are scientifically meaningful and interesting. We have developed an R

package (called statnet) to implement the procedures developed in this paper. The

package is available at http://csde.washington.edu/statnet.

It is possible to simulate random networks from a given ERGM — at least in prin-

ciple — using well-established Markov chain Monte Carlo techniques. More recently,

various researchers have been developing techniques to solve a harder problem: cal-

culating approximate maximum likelihood estimates of the ERGM parameters, given

an observed network. While these techniques are conceptually simple (Geyer and

Thompson 1992), their practical implementation for relatively large social networks

has proven elusive. We are now able to apply these techniques to networks encom-

passing thousands of nodes, problems much larger than those that could be tackled

until very recently.

In problems for which maximum likelihood estimation previously has been possible

in ERGMs, a troubling empirical fact has emerged: When ERGM parameters are es-

timated and a large number of networks are simulated from the resulting model, these

networks frequently bear little resemblance at all to the observed network (Handcock,

2003). This seemingly paradoxical fact arises because even though the maximum like-

lihood estimate makes the probability of the observed network as large as possible,

this probability might still be extremely small relative to other networks. In such a

case, the ERGM does not fit the data well.

The remainder of this article provides a case study illustrating the application

of recently developed models, software, and goodness of fit procedures to network

datasets from the National Longitudinal Study of Adolescent Health (AddHealth),

which is described in Section 2. Section 3 explains the statistical models we fit to

3

these data. Section 4 illustrates our goodness of fit technique on two simple models

that do not fit well. Section 5 explains a set of network statistics, which are then

used to build good-fitting models in Section 6.

2 Introduction to the AddHealth Survey

The network data on friendships that we study in this article were collected during

the first wave (1994–1995) of the National Longitudinal Study of Adolescent Health

(AddHealth). The AddHealth data come from a stratified sample of schools in the

US containing students in grades 7 through 12. To collect friendship network data,

AddHealth staff constructed a roster of all students in a school from school adminis-

trators. Students were then provided with the roster and asked to select up to five

close male friends and five close female friends. Students were allowed to nominate

friends who were outside the school or not on the roster, or to stop before nomi-

nating five friends of either sex. Complete details of this and subsequent waves of

the study can be found in Resnick et al. (1997) and Udry and Bearman (1998) and

at http://www.cpc.unc.edu/projects/addhealth. In most cases, the individual

school does not contain all grades 7–12; instead, data were collected from multiple

schools within a single system (e.g. a junior high school and a high school) to obtain

the full set of six grades. In these cases, we will use the term “school” to refer to a

set of schools from one community.

The full dataset contains 86 schools, 90,118 student questionnaires, and 578,594

friendship nominations. Schools with large amounts of missing data were excluded

from our analysis; this happened, among other reasons, for special education schools

and for school districts that required explicit parental consent for student participa-

tion. Thus, our analysis included 59 of the schools, ranging in size from 71 to 2209

surveyed students. However, in this article we focus primarily on a single illustrative

school, School 10, that has 205 students. Our results for School 10 may not necessar-

ily be inferred to the whole population of schools; in particular, as we point out in

Section 7, the parameter estimates for School 10 may be numerically quite different

for those of other schools because the parameters may depend on the number of nodes

4

in a complicated way. Yet when we consider all 59 of the schools, we find remarkably

similar qualitative results.

The edges in these raw network data are directed, since it is possible A could

name B as a friend without B nominating A. However, in this article we will consider

the undirected network of mutual friendships, those in which both A nominates B

and B nominates A. This feature of reciprocation of nomination is common to many

conceptualizations of friendship.

Each network may be represented by a symmetric n × n matrix Y and an n × q

matrix X of nodal covariates, where n is the number of nodes. The entries of the Y

matrix, termed the adjacency matrix, are all zeros and ones, with Yij = 1 indicating

the presence of an edge between i and j. Since self-nomination was disallowed, Yii = 0

for all i. The limit on the number of allowed nominations means that the data are

not complete, but we will assume for convenience that a lack of nomination in either

direction between two individuals means that there is no mutual friendship.

The nodal covariate matrix X includes many measurements on each of the indi-

viduals in these networks. Some such measurements, like sex, are not influenced by

network structure in any way, and are termed exogenous. Other covariates may ex-

hibit non-exogeneity: for example, tobacco use may be influenced through friendships.

Exogeneity is important, for instance, to guarantee the dyadic independence property

that we will explain in equation (3). We focus our analysis on only three covariates:

sex, grade, and race. Although the latter two may exhibit some endogeneity (e.g.,

the influence of friends may affect whether a student fails and must repeat a grade,

or which race a student of mixed-race heritage chooses to identify with), we assume

such effects are minimal and consider the attributes fixed and exogenous. What we

term “race” is constructed from two questions on race and Hispanic origin, with His-

panic origin taking precedence. Thus, our categories “Hispanic”, “Black”, “White”,

“Asian”, “Native American”, and “Other” are short-hand names for “Hispanic (all

races)”, “Black (non-Hispanic)”, “White (non-Hispanic)”, etc. This coding follows

standard practice in the social science literature.

5

3 Exponential Random Graph Models

Our overall goal in using exponential random graph models (ERGMs), also known as

p-star models (Wasserman and Pattison, 1996), is to model the random behavior of

the adjacency matrix Y, conditional on the covariate matrix X. Given a user-defined

p-vector g(Y,X) of statistics and letting η ∈ Rp denote the statistical parameter,

these models form a canonical exponential family (Lehmann, 1983),

Pη(Y = y|X) = κ−1 exp{ηtg(y,X)}, (1)

where the normalizing constant κ ≡ κ(η) is defined by

κ =∑w

exp{ηtg(w,X)} (2)

and the sum (2) is taken over the whole sample space of allowable networks w.

The objective in defining g(Y,X) is to choose statistics that summarize the social

structure of the network. The range of substantially motivated network statistics that

might be included in the g(Y,X) vector is vast — see Wasserman and Faust (1994)

for the most comprehensive treatment of these statistics. We will consider only a

few key statistics here, chosen to represent friendship selection rules that operate at

a local level. The goal is to test whether these local rules can reproduce the global

network patterns of clustering and geodesic distances (Morris, 2003).

Development of estimation methods for ERGMs has not kept pace with develop-

ment of ERGMs themselves. To understand why, consider the sum of equation (2).

A sample space consisting of all possible undirected networks on n nodes contains

2n(n−1)/2 elements, an astronomically large number even for moderate n. Therefore,

direct evaluation of the normalizing constant κ in equation (2) is computationally

infeasible for all but the smallest networks — except in certain special cases such

as the dyadic independence model of equation (3) — and inference using maximum

likelihood estimation is extremely difficult. To circumvent this difficulty, we use a

technique called Markov chain Monte Carlo maximum likelihood estimation in which

a stochastic approximation to the likelihood function is built and then maximized

(Geyer and Thompson 1992). This and other methods have been considered by

6

Dahmstrom and Dahmstrom (1993), Corander et al. (1998), Crouch et al. (1998),

Snijders (2002), and Handcock (2002). Details of the specific technique we use may

be found in Hunter and Handcock (2006), while a discussion of the background of

ERGMs in the networks literature may be found in Snijders (2002) or Hunter and

Handcock (2006).

An important special case of model (1) is the dyadic independence model, in which

g(y,X) =∑ ∑

i<j

yijh(Xi,Xj) (3)

for some function h mapping Rq × Rq into Rp, where the q-dimensional row vectors

Xi and Xj are the nodal covariate vectors for the ith and jth individuals. In the

context of an undirected network, the word dyad refers to a single Yij for some pair

(i, j) of nodes (not to be confused with an edge, which requires Yij = 1). In the

ERGM resulting from equation (3), equation (1) becomes

Pη(Y = y|X) = κ−1∏∏

i<j

exp{yijηt∆(g(y,X))ij}, (4)

where

∆(g(y,X))ij = g(y,X)|yij=1 − g(y,X)|yij=0 (5)

denotes the change in the vector of statistics when yij is changed from 0 to 1 and the

rest of y remains unchanged. In equation (4), the joint distribution of the Yij is simply

the product of the marginal distributions — hence the name “dyadic independence

model”. The MLE in such a model may be obtained using logistic regression. As the

simplest example of a dyadic independence model, we take p = 1 and h(Xi,Xj) =

1, which yields the well-known Bernoulli network, also known as the Erdos-Renyi

network, in which each dyad is an edge with probability exp{η}/(1 + exp{η}).For dyadic dependence models, equation (4) is not generally true, but nonetheless

the right hand side of this equation is called the pseudolikelihood. Until recently,

inference for social network models has relied on maximum pseudolikelihood esti-

mation, or MPLE, which may be implemented using a standard logistic regression

algorithm (Besag 1974; Frank and Strauss, 1986; Strauss and Ikeda, 1990; Geyer

7

and Thompson 1992). However, it has been argued that MPLE can perform very

badly in practice (Geyer and Thompson, 1992) and that its theoretical properties

are poorly understood (Handcock, 2003). Particularly dangerous is the practice of

interpreting standard errors from logistic regression output as though they are rea-

sonable estimates of the standard deviations of the pseudolikelihood estimators. The

only estimation technique we discuss for the remainder of this article is maximum

likelihood estimation.

4 Goodness of fit for dyadic independence models

The first dyadic independence model we consider is perhaps the simplest possible

network model, in which g(y,X) consists only of E(y), the number of edges in y.

This is the Bernoulli, or Erdos-Renyi, network described in Section 3. For AddHealth

school 10, the parameter estimate for the Bernoulli network is seen in Table 1 to be

−4.625. This may be derived exactly: Since school 10 has 205 nodes and 203 edges,

the MLE for the probability that any dyad has an edge is 203/(2052

), or 0.00971, and

the log-odds of this value is −4.625.

The second model we consider includes edges and also several statistics based on

nodal covariates. All of these statistics may be expressed as dyadic independence

statistics as in equation (3). That is, they are all of the form∑ ∑i<j

yijh(Xi,Xj) (6)

for a suitably chosen function h(Xi,Xj).

First, we include the so-called nodal factor effects for each of the factors grade,

race, and sex. Given a particular level of a particular factor (categorical variable),

the nodal factor effect counts the total number of endpoints with that level for each

edge in the network. In other words,

h(Xi,Xj) =

{2 if both nodes i and j have the specified factor level;1 if exactly one of i, j has the specified factor level;0 if neither i nor j has the specified factor level.

(7)

This means that the corresponding parameter is the change in conditional log-odds

when we add an edge with one endpoint having this factor level — and this change

8

is doubled when both endpoints of the edge share this level. As an example, consider

the grade factor, which has levels 7 through 12 along with one missing-value level NA.

These seven levels of the grade factor require six separate statistics for the nodal factor

effect; one level must be excluded since the sum of all seven equals twice the number

of edges in the network, thus creating a linear dependency among the statistics.

The second type of nodal statistics we employ are homophily statistics. A ho-

mophily statistic for a particular factor gives each edge in the network a score or zero

or one, depending on whether the two endpoints have matching values of the factor.

We distinguish between two kinds of homophily, depending on whether the distinct

levels of the factor should exhibit different homophily effects. Thus, for uniform

homophily, we have a single statistic, defined by

h(Xi,Xj) ={

1 if i and j have the same level of the factor;0 otherwise.

On the other hand, for differential homophily, we have a set of statistics, one for each

level of the factor, where each is defined by

h(Xi,Xj) ={

1 if i and j both have the specified factor level;0 otherwise.

(8)

Note that for sex, a two-level factor, we may include a differential homophily effect

or a nodal factor effect but not both. This is because in an undirected network, there

are only three types of edges — male-male, female-female, and male-female — so only

two statistics are required to completely characterize the sexes of both endpoints of

an edge, provided the overall edge effect is also in the model. A differential homophily

effect (two statistics) plus a nodal factor effect (one statistic) would together entail

redundant information.

In addition to the nodal factor and homophily effects, one final set of terms in

our second dyadic independence model (summarized as Model I in Table 2) involves

the grade factor. This is an ordinal categorical variable, and we may expect that

the propensity to form friendships depends on how different two individuals’ grade

values are (e.g., seventh graders may be more likely to form friendships with eighth

graders than twelfth graders). While one could add a new model term for each

possible pairing of two grade levels, a far more parsimonious model considers only

9

the absolute difference of grade values:

h(Xi,Xj) ={

1 if |gradei − gradej| = C for some constant C;0 otherwise.

(9)

In our models, we added terms according to equation (9) for C = 1, C = 2, and

C = 3. (We could not let C = 0, since this would introduce a linear dependence with

the homophily statistics.) This has the effect of combining C = 4 and C = 5, along

with any pairs for which grade is missing on one individual, into a single reference

category.

Note that all schools have two sexes and six grades, but only some have additional

NA categories for these factors. Furthermore, the number of races present varies

considerably from school to school. Parameters are excluded from the model when it

can be determined in advance that the MLE will be undefined. Such cases occur for

node factor effects when only a small number of students possess the factor level and

they all have 0 friendships; or for homophily terms, when there are no ties between

two students with a given factor level. For example, in AddHealth school 10, grade is

a seven-level factor, sex is a three-level factor, and race is a four-level factor; and our

dyadic independence model contains 25 parameters: one for edges, six for the grade

factor effect, six for differential homophily on grade (excluding the NA category), five

for the race factor effect, four for differential homophily on race (excluding the NA

and Other categories), two for the sex factor effect, and one for uniform homophily

on sex. The fitted values of these 25 parameters are presented as Model I in Table 2.

Our graphical tests of goodness-of-fit require a comparison of certain observed

network statistics with the values of these statistics for a large number of networks

simulated according to the fitted ERGM. The choice of these statistics determines

which structural aspects of the networks are important in assessing fit. We propose

to consider three sets of statistics: the degree distribution, the edgewise shared partner

distribution, and the geodesic distance distribution.

The degree distribution for a network consists of the values D0/n, . . . , Dn−1/n,

where Dk/n equals the proportion of nodes that share edges with exactly k other

nodes. The edgewise shared partner distribution consists of the values EP0/E, . . . , EPn−2/E,

where E denotes the total number of edges and EPk equals the number of edges whose

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endpoints both share edges with exactly k other nodes. (The Dk and EPk statistics

are explained in much greater detail in Section 5.) Finally, the geodesic distance

distribution consists of the relative frequencies of the possible values of geodesic dis-

tance between two nodes, where the geodesic distance between two nodes equals the

length of the shortest path joining those two nodes (or infinity if there is no such

path). For instance, because two nodes are at geodesic distance 1 if and only if they

are connected by an edge, and because there are(

n2

)possible pairs of nodes, the first

value of the geodesic distance distribution equals E/(

n2

). The last value, the fraction

of dyads with infinite geodesics, is also called the fraction “unreachable.”

We chose to include the degree statistics because of the tremendous amount of

attention paid to them in the networks literature. We included the shared partner

statistics based on the work of Snijders et al. (2006) and Hunter and Handcock (2006),

and because we will show (in Section 6) that the addition of a parametric formula

involving EP0, . . . , EPn−2 improves model fit dramatically. Therefore, these statistics

appear to contain a great deal of relevant network information. Furthermore, equation

(13) demonstrates that the triangle count, ubiquitous in the networks literature, is a

function of the shared partner statistics. Finally, the geodesic distance statistics are

the basis for two of the most common measures of centrality, a fundamental concept in

social network theory (Wasserman and Faust 1994, page 111), and are clearly relevant

to the speed and robustness of diffusion across networks. They also represent higher-

order network statistics not directly related to any of the statistics included in our

models, and thus provide a strong independent criterion for goodness of fit.

Figure 2 depicts the results of 100 simulations for School 10 from the fitted dyadic

independence models given in Tables 1 and 2. The vertical axis in each plot is the

logit (log-odds) of the relative frequency, and the solid line represents the statistics for

the observed network. We can immediately see that the models do an extremely poor

job of capturing the shared partner distribution. They perform relatively well for

the degree distribution and the geodesics distribution, considering their simplicity.

Adding the attribute-based statistics improves the fit of the geodesic distribution

considerably. The lack of fit in the shared partner plot reflects the fact that the

model strongly underestimates the amount of local clustering present in the data.

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(a) School 10, edges only (Bernoulli or Erdos-Renyi model)

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Figure 2: Simulation results for dyadic independence models. In all plots, the verticalaxis is the logit of relative frequency; the School 10 statistics are indicated by thesolid lines; the boxplots include the median and interquartile range; and the lightgray lines represent the range in which 95 percent of simulated observations fall.

The models predict friends to have no friends in common most of the time, and

occasionally one friend in common, whereas in the original data they have up to five.

Although we present plots for only one school here, the qualitative results for other

schools follow a small number of similar patterns.

In Sections 5 and 6, we present some modifications to the models seen here that fit

much better as measured both by the graphical criterion we have employed here and

by more traditional statistical measures such as Akaike’s Information Criterion (AIC).

The fact that the simple dyadic independence models do not appear to fit the data

well is not surprising; after all, such models are merely logistic regression models in

which the responses are the dyads. That we must move beyond dyadic independence

in order to construct models that fit social network data well is a result of the fact

that the formation of edges in a network depends upon the existing network structure

12

itself.

5 Degree, shared partner, and other network statis-

tics

A simplistic ERGM that is not a dyadic independence model is one in which g(y,X)

consists only of a subset of the degree statistics Dk(y), 0 ≤ k ≤ n − 1. The degree

of a node in a network is the number of neighbors it has, where a neighbor is a

node with which it shares an edge. We define Dk(y) to be the number of nodes

in the network y that have degree k. Note that the Dk(y) statistics satisfy the

constraint∑n−1

i=0 Di(y) = n, so we may not include all n degree statistics among the

components of the vector g(y,X); if we did, the coefficients in model (1) would not

be identifiable. A common reformulation of the degree statistics is given by the k-star

statistics S1(y), . . . , Sn−1(y), where Sk(y) is the number of k-stars in the network y.

A k-star (Frank and Strauss, 1986) is an unordered set of k edges that all share a

common node. For instance, “1-star” is synonymous with “edge”. Since a node with

i neighbors is the center of(

ik

)k-stars (but the “common node” of a 1-star may be

considered arbitrarily to be either of two nodes), we see that

Sk(y) =n−1∑i=k

(i

k

)Di(y), 2 ≤ k ≤ n− 1; and S1(y) =

1

2

n−1∑i=1

i Di(y). (10)

Note that an edge is the same as a 1-star, so E(y) = S1(y). The k-star statistics are

highly collinear with one another. For example, any 4-star automatically comprises

four 3-stars, six 2-stars, and four 1-stars (or edges).

The shared partner statistics are another useful class of statistics. We define two

distinct sets of shared partner statistics, the edgewise shared partner statistics and

the dyadwise shared partner statistics. The edgewise shared partner statistics are

denoted EP0(y), . . . , EPn−2(y), where EPk(y) is defined as the number of unordered

pairs {i, j} such that yij = 1 and i and j have exactly k common neighbors (Hunter

and Handcock, 2006). The requirement that yij = 1 distinguishes the edgewise shared

partner statistics from the dyadwise shared partner statistics DP0(y), . . . , DPn−2(y):

We define DPk(y) to be the number of pairs {i, j} such that i and j have exactly k

13

common neighbors. In particular, it is always true that DPk(y) ≥ EPk(y), and in

fact DPk(y)− EPk(y) equals the number of unordered pairs {i, j} for which yij = 0

and i and j share exactly k common neighbors.

Since there are E(y) edges and(

n2

)dyads in the entire network, we obtain the

identities

E(y) =n−2∑i=0

EPi(y) (11)

and (n

2

)=

n−2∑i=0

DPi(y). (12)

Furthermore, we can obtain the number of triangles in y by considering the edgewise

shared partner statistics: Whenever yij = 1, the number of triangles that include this

edge is exactly the number of common neighbors shared by i and j. Therefore, if

we count all of the shared partners for all edges, we have counted each triangle three

times, once for each of its edges. In other words,

T(y) =1

3

n−2∑i=0

i EPi(y). (13)

A related formula involving the dyadwise shared partner statistics is obtained by

noting that each triangle automatically comprises three 2-stars. Therefore, S2(y) −3T(y) is the number of 2-stars for which the third side of the triangle is missing. We

conclude that

S2(y)− 3T(y) =n−2∑i=0

i [DPi(y)− EPi(y)] . (14)

Combining equation (14) with equation (13) produces

S2(y) =n−2∑i=0

i DPi(y).

Because a 2-star is also a path of length two, S2(y) is sometimes referred to as the

twopath statistic.

14

12

34

5

Figure 3: For this simple five-node network, the edgewise and dyadwise shared part-ner distributions are (EP0, . . . , EP3) = (1, 4, 1, 0) and (DP0, . . . , DP3) = (2, 6, 2, 0),respectively; the k-triangle and k-twopath distributions are (T1, T2, T3) = (2, 1, 0)and (P1, P2, P3) = (10, 1, 0), respectively.

Finally, we summarize two additional sets of statistics, due to Snijders et al.

(2006), that will be used in Section 6. First, the triangle statistic generalizes to

the set of k-triangle statistics, where a k-triangle is defined to be a set of k distinct

triangles that share a common edge. In particular, a 1-triangle is the same thing as a

triangle. Second, the 2-star statistic (also known as the twopath statistic) generalizes

to the set of k-twopath statistics, where a k-twopath is a set of k distinct 2-paths

joining the same pair of nodes. In particular, a 1-twopath is the same thing as a 2-star

or a 2-path. Snijders et al (2006) actually coined the term “k-independent 2-path,”

but we simplify this to k-twopath in this article.

As a concrete example, we note that in the simple network of Figure 3, there are

two 1-triangles, one 2-triangle, ten 1-twopaths, and one 2-twopath. (Note that the

2-twopath joining nodes 1 and 4 is the same as the 2-twopath joining nodes 2 and 3,

though it is counted only once.) We denote the number of k-triangles and k-twopaths

in the network y by Tk(y) and Pk(y), respectively. Just as the degree statistics Di(y)

are related to the k-star statistics Sk(y) by (10), the edgewise and dyadwise shared

partner statistics are related to the k-triangle and k-twopath statistics, respectively,

by the equations

Tk(y) =n−2∑i=k

(i

k

)EPi(y), 2 ≤ k ≤ n− 2

and

Pk(y) =n−2∑i=k

(i

k

)DPi(y), 1 ≤ k ≤ n− 2, k 6= 2.

15

The cases not covered above are that of T1(y), given in equation (13), and P2(y), the

number of 4-cycles, which includes an extra factor of 1/2 because any 4-cycle can be

considered a 2-path between two distinct pairs of nodes:

P2(y) =1

2

n−2∑i=2

(i

2

)DPi(y).

6 Goodness of fit for dyadic dependence models

A fundamental principle of social network analysis is that dependence among edges

is a guiding force in the formation of networks. That is, the Yij that make up an

adjacency matrix Y are not independent of one another. For instance, it seems

reasonable that two potential edges should be correlated if they involve the same

individual (so that Yij is not independent of Yik for distinct i, j, and k). Starting

from this simple assumption, Frank and Strauss (1986) proposed the so-called Markov

random graphs. They showed that an ERGM with a few simple network statistics,

among them the number of triangles, suffices under this assumption. As originally

proposed, these homogeneous Markov random graphs treated all nodes as equivalent,

ignoring any covariate information.

Yet what if we have covariate information about the nodes? Since nodal attributes

are clearly important in most, if not all, social networks, it is not surprising that

homogeneous Markov random graph models have failed empirically to describe social

network data. But is it true that the Yij are dependent even when we condition

on the observations X? One might take the approach suggested by Markov random

graphs, adding covariate information of the sort discussed in Section 4 to Markov

random graph models and then performing statistical tests to determine whether

terms such as the number of triangles are statistically significant after the effects

of covariates are accounted for. Unfortunately, this approach fails. The reasons for

this failure are deep (see Handcock, 2002 and Handcock, 2003), but in a nutshell,

the models obtained by adding covariate information to Markov-inspired ERGMs

simply do not fit network data well. Without a model that yields simulated networks

resembling the observed network, maximum likelihood estimation itself is doomed

16

(for a discussion of maximum likelihood estimation algorithms, see Snijders, 2002

or Hunter and Handcock, 2006). This failure motivated the work of Snijders et al.

(2006) in developing the alternating k-triangle, k-twopath, and k-star statistics that

we explain in this section.

These new statistics finally allow us to build dependence models that fit net-

work datasets well enough to enable reliable maximum likelihood estimation rou-

tines. Thus, we can augment these models by adding covariate-only terms, then try

to determine whether the dependence terms are statistically significant. If not, then

independence models will suffice, so one may simply perform “network” analysis by

ignoring the network structure altogether and performing logistic regression on the

independent responses Yij. We will see (somewhat reassuringly from the point of view

of social networks research) that the dependence among dyads appears to persist even

after nodal covariate information is taken into account.

Consider using the shared partner statistics and the degree statistics defined in

Section 5 to build an ERGM. For instance, it is possible to add one new term to the

model for each of the edgewise shared partner statistics EP1, . . . , EPn−2 — we omit

EP0 to avoid the linear dependence of equation (11) — but this typically leads to

a model with too much flexibility. As Hunter and Handcock (2006) point out, it is

often better to restrict the parameter space to avoid problems of degeneracy. To this

end, we define the statistics

u(y; τ) = eτ

n−2∑i=1

{1−

(1− e−τ

)i}

Di(y), (15)

v(y; τ) = eτ

n−2∑i=1

{1−

(1− e−τ

)i}

EPi(y), (16)

and w(y; τ) = eτ

n−2∑i=1

{1−

(1− e−τ

)i}

DPi(y), (17)

where τ in each case is an additional parameter. We refer to these three statistics as

geometrically weighted degree, edgewise shared partner, and dyadwise shared partner

statistics, respectively.

Although the definitions of u, v, and w may appear somewhat unusual, they are

chosen to coincide with the alternating k-star, alternating k-triangle, and alternating

17

k-twopath statistics, respectively, of Snijders et al. (2006):

u(y; τ) = 2S1(y)− S2(y)

(eτ )1+ · · ·+ (−1)n Sn−1(y)

(eτ )n−2, (18)

v(y; τ) = 3T1(y)− T2(y)

(eτ )1+ · · ·+ (−1)n−3 Tn−2(y)

(eτ )n−3, (19)

and w(y; τ) = P1(y)− 2P2(y)

(eτ )1+ · · ·+ (−1)n−3 Pn−2(y)

(eτ )n−3. (20)

[Actually, the alternating k-star statistic in equation (18) is not identical to the al-

ternating k-star statistic of Snijders et al. (2006), though it is in a certain sense

equivalent from a modeling perspective. Hunter (2007) discusses this issue at length.]

As Snijders et al. (2006) explain, these three statistics appear to capture high-order

dependency structure in networks in a parsimonious fashion while avoiding the prob-

lems of degeneracy described by Handcock (2002; 2003).

The τ parameters in equations (16), (17), and (15) are not canonical exponen-

tial family parameters like η in equation (1); rather, if τ = (τ1, τ2, τ3) is considered

unknown and (η, τ ) is the full parameter vector, then the ERGM forms a curved

exponential family, which complicates the estimation procedure. Hunter and Hand-

cock (2006) address this more complicated situation; however, for the purposes of this

article, we make the simplifying assumption that each τ is fixed and known. In our

model-fitting procedure, we tried a range of different values of τ on several schools

and found that for each statistic, the goodness-of-fit plots (as in Figure 4) were nearly

indistinguishable for different values of τ in the range we tested (0.1 to 1.5). Values

far outside this range resulted in models that could not be fit. Based on these results,

we use a fixed value of τ = 0.25 for all the models we discuss below.

As an example, we take g(y,X) to consist of only two terms, the edge statistic

and the geometrically weighted edgewise shared partner (GWESP) statistic. In this

case, the ERGM of equation (1) becomes

Pη(Y = y|X) = κ−1 exp{η1E(y) + η2v(y; τ)}. (21)

We fit model (21), as well as similar models using the geometrically weighted dyadwise

shared partner (GWDSP) and geometrically weighted degree (GWD) statistics, to

AddHealth school 10. The results are found in Table 1.

18

Model:Coefficient: Edges only Edges plus GWESP Edges plus GWDSP Edges plus GWDedges −4.625(0.07)∗∗∗ −5.280(0.10)∗∗∗ −4.812(0.197)∗∗∗ −1.423(0.50)∗∗

GWESP 1.544(0.10)∗∗∗

GWDSP 0.047(0.046)GWD −1.305(0.20)∗∗∗

∗∗ Significant at 0.01 level ∗∗∗ Significant at 0.001 level

Table 1: Estimated coefficients and standard errors for the parameters of three simplemodels that consider only network structure but no nodal covariate information.The GWESP statistic v(y; τ), the GWDSP statistic w(y; τ), and the GWD statisticu(y; τ) all use τ = 0.25.

Many dyadic dependence models create such severe numerical difficulties in esti-

mation (Handcock, 2002; 2003) that we are unable to fit them successfully for a large

number of different networks of different sizes. However, models with the GWESP,

GWDSP, and GWD statistics appear to be more robust: Using our MCMC fitting

procedure, we were able to be estimate their parameters on many of the AddHealth

schools, the first such application of maximum likelihood estimation to a dyadic de-

pendence model for a range of different-sized networks with hundreds of nodes. As

a case in point, consider Figure 6, in which we successfully fit a dyadic dependence

model to the largest school in the sample, with 2209 nodes, and obtained reasonable

parameter estimates. (We discuss this school further in Section 7.)

As described in Section 4, one way to develop an idea of how well a model fits is

by comparing a set of observed network statistics with the range of the same statistics

obtained by simulating many networks from the fitted ERGM. If the observed network

is not typical of the simulated network for a particular statistic, then the model is

either degenerate (if the statistic is among those included in the ERGM vector g[y,X])

or poorly-fitting (if the statistic is not included). Figure 4 depicts simulation results

for school 10 for the three dyadic-dependent ERGMs in Table 1; Figure 5 depicts

Model II from Table 2.

For both School 10 and many of the other smaller AddHealth schools, a simple

model containing only individual-level attributes [Figure 2(b)] does a respectable

19

Coefficient Model I Model II Coefficient Model I Model IIedges −10.45(1.44)∗∗∗ −3.49(1.92) AD (Gr.) = 1 3.71(1.26)∗∗ 3.41(1.42)∗

GWESP 0.83(0.13)∗∗∗ AD (Gr.) = 2 2.43(1.28) 2.42(1.48)GWD −2.01(0.35)∗∗∗ AD (Gr.) = 3 1.52(1.42) 1.43(1.62)GWDSP 0.50(0.09)∗∗∗

DH (Gr. 7) 6.83(1.37)∗∗∗ 6.00(1.56)∗∗∗

NF (Gr. 8) −0.54(0.73) −0.34(0.78) DH (Gr. 8) 7.81(1.56)∗∗∗ 6.48(1.64)∗∗∗

NF (Gr. 9) 0.72(0.53) 0.64(0.59) DH (Gr. 9) 5.04(1.39)∗∗∗ 4.52(1.58)∗∗

NF (Gr. 10) 0.59(0.57) 0.55(0.59) DH (Gr. 10) 5.76(1.44)∗∗∗ 4.96(1.59)∗∗

NF (Gr. 11) 1.12(0.53)∗ 0.97(0.60) DH (Gr. 11) 4.98(1.39)∗∗∗ 4.32(1.54)∗∗

NF (Gr. 12) 1.51(0.60)∗ 1.23(0.60)∗ DH (Gr. 12) 4.64(1.48)∗∗ 4.11(1.58)∗∗

NF (Gr. NA) 4.05(1.14)∗∗∗ 3.86(1.30)∗∗

DH (White) 1.58(0.62)∗ 1.55(0.68)∗

NF (Black) 0.45(0.39) 0.51(0.42) DH (Black) 1.18(1.27) 0.92(1.55)NF (Hisp) −0.419(0.34) −0.23(0.33) DH (Hisp) 1.17(0.41)∗∗ 0.87(0.43)∗

NF (Nat Am) −0.460(0.30) −0.21(0.32) DH (Nat Am) 1.70(0.42)∗∗∗ 1.31(0.43)∗∗

NF (Other) −1.14(0.75) −0.61(0.69)NF (Race NA) 1.23(0.61)∗ 1.53(0.89)

NF (Female) 0.09(0.09) 0.09(0.10) UH (Sex) 0.78(0.15)∗∗∗ 0.67(0.16)∗∗∗

NF (Sex NA) −0.43(0.47) −0.18(0.47)NF stands for Node Factor. AD stands for Absolute Difference.

DH stands for Differential Homophily.UH stands for Uniform Homophily.

∗ Significant at 0.05 level ∗∗ Significant at 0.01 level ∗∗∗ Significant at 0.001 level

Table 2: Estimated coefficients (and standard errors) for two models applied to Ad-dHealth school 10. Model I contains terms for edges and the 25 nodal covariate termsdescribed in Section 4. Model II contains all of the terms in Model I plus threeadditional terms, GWESP, GWDSP, and GWD, each with τ = 0.25. Differential ho-mophily terms for Grade NA, Race Other, Race NA, and Sex NA are omitted becausethere are no edges observed between two actors sharing these attribute values.

20

(a) School 10, edges and GWESP (τ = 0.25)

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job of recreating the geodesic distribution of the observed data, a global property

of the network. At the same time, it strongly underestimates the amount of local

clustering as captured by the shared partner distribution. The former observation

is encouraging, since information on attribute matching is far easier to collect than

other types of network data in most real-world settings where only a sample of nodes

is available: Gathering such information only requires questions about the attributes

of respondents’ partners, not their actual identities. The latter observation tells us

21

School 10, edges, covariates, GWESP, GWDSP, and GWD (all τ = 0.25)

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Figure 5: Simulation results for Model II of Table 2.

that not all features of the network can be ascribed to purely dyadic-level phenomena

— yet this fact is not surprising, as it is the very basis for the field of network analysis.

Finally, the fact that a simple model is strongly predictive of one higher-order network

property (geodesics) and strongly divergent from another (shared partner) suggests

that a variety of network statistics ought to be tested in order to develop a robust

sense of goodness-of-fit.

Comparing Figure 2(b) with Figures 4(a) and 4(b), we see that incorporating the

heterogeneity of actors through nodal covariates was more important for model fit

than either modelling degree or edgewise shared partners alone. This should not be

too surprising; we expect that nodal covariates are very important in predicting most

types of social relationships, and certainly high school friendships are no exception.

Social relations generally exhibit local clustering, and in this case we observe that

the simple Bernoulli model drastically under-predicts the number of shared partners

people should have, even though it captures the degree distribution well. Such cluster-

ing can come from at least two different sources: (1) actors matching on exogenous at-

tributes; and (2) actors forming partnerships on the basis of existing shared partners.

The two are fundamentally different: the former is dyadic-independent, using factors

exogenous to the network structure; while the latter is dyadic-dependent and reflects

the transitivity property that friends of my friends are more likely to be my friends.

The modelling here shows that neither homophily nor shared partners alone is suffi-

cient to explain the clustering observed in this friendship network (the same is true of

22

other AddHealth schools; see the plots at http://csde.washington.edu/networks).

Indeed, in Table 2 we see that the homophily effects are smaller in magnitude in Model

II, which includes the shared partner statistics, than in Model I.

In this setting, a simple one-term Bernoulli model [Figure 2(a)] turned out to

fit the degree distribution fairly well, though it fails slightly in underestimating the

number of high-degree nodes. Adding the degree distribution to the model in the form

of the GWD statistic [Figure 4(c)] appears to remedy this defect. However, neither

of these two models reproduced the clustering and geodesic distances observed in this

network. In fact, comparing the rightmost plots in Figures 2(a) and 4(c), we see that

adding the GWD statistic by itself appears to have worsened the fit to the geodesic

distance distribution. We might explain this phenomenon by noting (in Figure 1) that

the students tend to form friendships preferentially within their own grade, creating

longer-than-expected geodesics between pairs of students in different grades. Adding

only the GWD term to the model makes this problem worse because it tends to create

highly-connected individuals (while still ignoring grade), thus shortening the geodesics

even more. But whatever the cause of this phenomenon, the most important message

here is that there are hazards inherent in focusing only on the degree distribution

when trying to build realistic models for networks.

One might wonder how our graphical methods for assessing goodness of fit compare

with more traditional methods such as AIC (Akaike, 1973) or BIC (Schwarz, 1978),

where

AIC(M) = −2(maximized loglikelihood under M) + 2(# of parameters in M) (22)

and

BIC(M) = −2(maximized loglikelihood under M) + log N(# of parameters in M)(23)

where M denotes a particular ERGM and N denotes the sample size. The goal

is to minimize AIC(M) or BIC(M) as a function of M . Unfortunately, there are

several problems with these traditional methods. For one thing, the assumptions

used to justify AIC and BIC are not met here because our observations are not an

independent and identically distributed sample. In fact, it is not even clear how to

23

evaluate BIC(M) because there is no easy way to determine the effective sample size

N . For any dyadic independence model, N =(

n2

), the number of dyads. However,

when dependence among dyads exists, the effective sample size can be smaller than(n2

). Finally, as we pointed out in Section 3, it is not possible to evaluate the likelihood

function directly for most ERGMs except in the case of dyadic independence models,

where the likelihood equals the pseudolikelihood (4).

Thus, any attempt to achieve model selection via AIC or BIC is approximate at

best. However, even using the roughly approximated AIC, we find that the conclusions

of the graphical goodness-of-fit procedures are borne out in the sense that models that

produce large reductions in (approximate) AIC also seem to yield considerably better

fits in the graphical plots; those with smaller reductions in AIC have less pronounced

effects on the plots. However, the goodness of fit plots provide a richer picture than

AIC alone. From these plots, a number of features of the relationships between these

models and the network structure become clear. For instance, both the plots and

AIC indicate that incorporating the heterogeneity of actors through nodal covariates

is far more important for model fit than modelling either degree or shared partners

alone. Yet the plots are more informative than the AIC results in the sense that they

tell which structural features are fit well and which are not.

7 Discussion

Only with the recent development of the R package statnet, which is available on the

internet at http://csde.washington.edu/statnet, has it been possible to reliably

apply likelihood-based inference for exponential random graph models (ERGMs) to

networks of hundreds of nodes. It is therefore possible to consider aspects of these

models such as how well they fit the observed data, as we do in this article for the

particular case of high school friendship networks.

Choosing an appropriate set of network statistics on which to compare the ob-

served network with networks simulated from the fitted model is an important task

in the graphical goodness-of-fit studies we advocate in this article. If possible, these

statistics should match the purpose for which one is estimating and simulating net-

24

works. It may not be immediately clear what kinds of network properties are relevant;

in fact, that might be precisely the question in which we are interested in the first

place. For many social relations, theory may suggest that people do not look be-

yond more than one or two layers of network neighbors, so adequately modelling

statistics such as the edgewise shared partner distribution might be expected to get

higher-order statistics correct as well.

When we compare different AddHealth schools, we find that many significant

model parameters show remarkably similar qualitative patterns. Even the numerical

values of the maximum likelihood estimates are often quite similar across friendship

networks. However, it is important when comparing networks with different numbers

of nodes that the values of the parameter estimates are not necessarily comparable.

The question of how to modify ERGMs so that their coefficients are directly compa-

rable without regard to n, the number of nodes, is a very important issue in network

modelling. Furthermore, as we pointed out in Section 6, the related question of the

effective sample size of a network on n nodes for a particular ERGM is important

if we have any hope of applying model selection methods such as BIC that depend

on sample size. However, this is a question for the future; for now, the science of

likelihood-based methods for fitting ERGMs is still in its early stages.

Although the most complete and best-fitting model presented here appears to

come close to capturing the higher-order network statistics examined for School 10

and many of the smaller schools, the same is not true for many of the larger schools.

For instance, consider Figure 6, based on the largest school in our sample, with 2209

nodes. This and other large schools depart from the fitted model in a similar way:

The model under-predicts the number of long geodesics and over-predicts the number

of short ones. In effect, the real social networks are more ”stringy” than our best-

fitting model predicts. See Goodreau (2007) for a more detailed analysis of a large

school.

As this empirical application has shown, both exogenous nodal covariates and

endogenous network effects can play an important role in the generative processes that

give rise to network structure. There is no a priori reason to assume that all networks

will have the same structure, and the methods here provide a systematic framework

25

School 44, edges, covariates, GWESP, GWDSP, and GWD (all τ = 0.25)

0 1 2 3 4 5 6 7 8 9 10 12

−7

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de

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Figure 6: Goodness-of-fit plots for the largest AddHealth school, school 44, with 2209nodes. The clear lack of fit in the geodesic distribution is typical of this model for thelarger AddHealth schools, even though the same model tends to fit well on smallerschools.

for evaluation of models that can be adapted to test a wide range of hypotheses. In the

context of mutual friendships among high-school adolescents, geometrically weighted

degree, edgewise shared partner, and dyadwise shared partner statistics — equivalent

to the alternating k-star, k-triangle, and k-twopath statistics, respectively, of Snijders

et al. (2006) — do a credible job of capturing the aggregate network structures of

interest.

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